Solve for $x$ : $ 8|x - 2| - 5 = 5|x - 2| + 3 $
Answer: Subtract $ {5|x - 2|} $ from both sides: $ \begin{eqnarray} 8|x - 2| - 5 &=& 5|x - 2| + 3 \\ \\ { - 5|x - 2|} && { - 5|x - 2|} \\ \\ 3|x - 2| - 5 &=& 3 \end{eqnarray} $ Add ${5}$ to both sides: $ \begin{eqnarray} 3|x - 2| - 5 &=& 3 \\ \\ { + 5} &=& { + 5} \\ \\ 3|x - 2| &=& 8 \end{eqnarray} $ Divide both sides by ${3}$ $ \dfrac{3|x - 2|} {{3}} = \dfrac{8} {{3}} $ Simplify: $ |x - 2| = \dfrac{8}{3}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 2 = -\dfrac{8}{3} $ or $ x - 2 = \dfrac{8}{3} $ Solve for the solution where $x - 2$ is negative: $ x - 2 = -\dfrac{8}{3} $ Add ${2}$ to both sides: $ \begin{eqnarray} x - 2 &=& -\dfrac{8}{3} \\ \\ {+ 2} && {+ 2} \\ \\ x &=& -\dfrac{8}{3} + 2 \end{eqnarray} $ Change the ${ + 2}$ to an equivalent fraction with a denominator of $3$ $ x = - \dfrac{8}{3} {+ \dfrac{6}{3}} $ $ x = -\dfrac{2}{3} $ Then calculate the solution where $x - 2$ is positive: $ x - 2 = \dfrac{8}{3} $ Add ${2}$ to both sides: $ \begin{eqnarray} x - 2 &=& \dfrac{8}{3} \\ \\ {+ 2} && {+ 2} \\ \\ x &=& \dfrac{8}{3} + 2 \end{eqnarray} $ Change the ${ + 2}$ to an equivalent fraction with a denominator of $3$ $ x = \dfrac{8}{3} {+ \dfrac{6}{3}} $ $ x = \dfrac{14}{3} $ Thus, the correct answer is $x = -\dfrac{2}{3} $ or $x = \dfrac{14}{3} $.